Theorem spw 2054

Description: Weak version of the specialization scheme sp 2218. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2218 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2218 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2169 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2218 are spfw 2053 (minimal distinct variable requirements), spnfw 1999 (when 𝑥 is not free in ¬ 𝜑), spvw 2001 (when 𝑥 does not appear in 𝜑), sptruw 1826 (when 𝜑 is true), spfalw 2000 (when 𝜑 is false), and spvv 2008 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1930	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1930	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1930	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2053	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800

This theorem is referenced by:  hba1w  2069  19.8aw  2072  exexw  2073  spaev  2074  ax12w  2167  rspw  3239  reldisj  4407  ralidmw  4470  dtruALT2  5327  bj-ssblem1  37126  bj-ax12w  37150  eu6w  43258